A certain formal power series attached to local densities of quadratic forms. II
We use the same notation as in the review above. Let \(B_ 1, \dots, B_ n\), \(A\) be symmetric regular matrices with entries in \(\mathbb{Z}_ p\). We now define another power series as follows \[ R \bigl( (B_ 1, \dots, B_ n), A; x_ 1,\dots, x_ n \bigr)= \sum_{e_ 1\geq \cdots\geq e_ n\geq 0} \alpha_ p \bigl( \langle p^{e_ 1} B_ 1, \dots, p^{e_ n} B_ n\rangle, A\bigr) x_ 1^{e_ 1} \cdots x_ n^{e_ n}. \] Suppose \(A\) is an unimodular matrix whose diagonal components belong to \(2\mathbb{Z}_ p\). Then Theorem 1 gives a recursion formula for \(\alpha_ p (\langle p^{2e} B_ 1, B_ 2\rangle, A)\) for diagonal matrices \(B_ 1\), \(B_ 2\) and large \(e\). Using Theorem 1 we obtain that for certain diagonal matrices \(B_ 1,\dots, B_ n\) the power series \(R((B_ 1, \dots, B_ n),A; x_ 1, \dots, x_ n)\) is a rational function in \(x_ 1,\dots, x_ n\) and we get a description of a denominator. Proofs of these results are announced to appear elsewhere.
scientific article; zbMATH DE number 701484